physik_iii
TRANSCRIPT
-
8/7/2019 physik_iii
1/58
Physics III
Summary
Prof. Dr. Simon Lilly
Autor
Thomas Gersdorf
ETH Zurich, HS 2008
$Id: physik_iii.tex 800 2008-12-07 20:46:56Z [email protected] $
Comitted: Monday 8th December, 2008 Revision 32
-
8/7/2019 physik_iii
2/58
Contents
Contents 2
1 Basics 7
1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Constants and Units . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Concept of charge . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Electrostatics 9
1 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1 Coulombs Law . . . . . . . . . . . . . . . . . . . . . . . 91.2 Energy of a charge . . . . . . . . . . . . . . . . . . . . . 9
2 Eletric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Charge density . . . . . . . . . . . . . . . . . . . . . . . 102.2 Curl of the electric field . . . . . . . . . . . . . . . . . . 11
3 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1 Flux and E-Fields of various charge distributions . . . 11
4 Gauss Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1 integral form . . . . . . . . . . . . . . . . . . . . . . . . 124.2 differential form . . . . . . . . . . . . . . . . . . . . . . 12
5 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.1 Definiton of potential 0 . . . . . . . . . . . . . . . . . . 135.2 Potential of radial-symmetrical charge distribution . . 13
5.3 Potential of multiple charges . . . . . . . . . . . . . . . 135.4 Potential of a spherical shell . . . . . . . . . . . . . . . 145.5 Potential of an electric dipole . . . . . . . . . . . . . . . 14
6 Energy of an electric field . . . . . . . . . . . . . . . . . . . . . 14
3 Conductors 15
1 Conductors and Insulators . . . . . . . . . . . . . . . . . . . . . 152 General Electrostatic Problem . . . . . . . . . . . . . . . . . . . 153 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . 154 Potential and Field inside an empty cavity in a closed conductor 16
2
-
8/7/2019 physik_iii
3/58
5 Mirror principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 conducting sphere in external field . . . . . . . . . . . . . . . . 167 Capicatance and capacitors . . . . . . . . . . . . . . . . . . . . 17
7.1 Definition of capicatance . . . . . . . . . . . . . . . . . 177.2 Types of capacitors . . . . . . . . . . . . . . . . . . . . . 177.3 Energy stored in a capicator . . . . . . . . . . . . . . . 187.4 Systems of capicators . . . . . . . . . . . . . . . . . . . 18
4 Currents 19
1 Electric currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.1 Definition of the current . . . . . . . . . . . . . . . . . . 19
2 Ohms law and Resistance . . . . . . . . . . . . . . . . . . . . . 20
2.1 Resistance and conductivity . . . . . . . . . . . . . . . . 202.2 Definiton of the resistance . . . . . . . . . . . . . . . . . 20
3 Resistance for various components . . . . . . . . . . . . . . . . 213.1 Discrete component . . . . . . . . . . . . . . . . . . . . 213.2 Spherical conductors . . . . . . . . . . . . . . . . . . . . 213.3 Cylindrical conductiors . . . . . . . . . . . . . . . . . . 21
4 Resistances in circuits . . . . . . . . . . . . . . . . . . . . . . . 214.1 Parallel circuits . . . . . . . . . . . . . . . . . . . . . . . 214.2 Series circuits . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Combined series and parallel circuits . . . . . . . . . . 22
5 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 Sources of electromotive force: Voltaic cell . . . . . . . . . . . 237 Circuits with capicators . . . . . . . . . . . . . . . . . . . . . . 23
5 Fields of moving charges 24
1 General force on a moving charge . . . . . . . . . . . . . . . . 242 Charge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Fields of moving charges . . . . . . . . . . . . . . . . . . . . . . 244 Electric field for an accelarated charge . . . . . . . . . . . . . . 245 Forces between moving charges - the magnetic force . . . . . 25
6 Magnetic fields 26
1 Definition of magnetic field . . . . . . . . . . . . . . . . . . . . 261.1 Lorentz force . . . . . . . . . . . . . . . . . . . . . . . . 261.2 Definition of B-field . . . . . . . . . . . . . . . . . . . . 26
2 Force between two wires . . . . . . . . . . . . . . . . . . . . . . 273 Properties of the magnetic field . . . . . . . . . . . . . . . . . . 27
3.1 Amperes Law . . . . . . . . . . . . . . . . . . . . . . . 274 Magntic poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . 286 Vector potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1 Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3
-
8/7/2019 physik_iii
4/58
Contents
6.2 General formula . . . . . . . . . . . . . . . . . . . . . . 29
6.3 Application to a current carrying wire . . . . . . . . . 296.4 General differentials . . . . . . . . . . . . . . . . . . . . 30
7 Fields of rings and solenoids . . . . . . . . . . . . . . . . . . . 308 Change in B across a current sheet . . . . . . . . . . . . . . . . 31
8.1 Energy density . . . . . . . . . . . . . . . . . . . . . . . 319 Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210 How do E and B transform together? . . . . . . . . . . . . . . 33
10.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2 General transformation . . . . . . . . . . . . . . . . . . 34
7 Magnetic Induction 35
1 Magnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . 351.1 Isolated conductor . . . . . . . . . . . . . . . . . . . . . 351.2 Loop of wire . . . . . . . . . . . . . . . . . . . . . . . . 35
2 EMF - Electro motive force . . . . . . . . . . . . . . . . . . . . 362.1 General emf (electro-motive force) . . . . . . . . . . . . 36
3 Magnetic flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Faradays law of induction . . . . . . . . . . . . . . . . . . . . . 37
4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Lenz Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Mutual inductance . . . . . . . . . . . . . . . . . . . . . . . . . 387 Self inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.1 Circuits with self-inductance . . . . . . . . . . . . . . . 398 Energy in a magnetic field . . . . . . . . . . . . . . . . . . . . . 40
8.1 General term for any B-field . . . . . . . . . . . . . . . 408.2 Derivation of the energy stored in a solenoid . . . . . . 40
9 Application: Magnetic field brake . . . . . . . . . . . . . . . . 40
8 Alternating Currents 42
1 Definitions for AC-circuits . . . . . . . . . . . . . . . . . . . . . 421.1 Quality factor . . . . . . . . . . . . . . . . . . . . . . . . 42
2 RCL circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 over-damped RCL-circuit . . . . . . . . . . . . . . . . . . . . . 43
4 RL-Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 RC-circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 RLC-circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Application: Media player . . . . . . . . . . . . . . . . . . . . . 458 AC currents and voltages as complex numbers . . . . . . . . . 459 Power consumptation in AC current . . . . . . . . . . . . . . . 4710 Some other definition . . . . . . . . . . . . . . . . . . . . . . . . 48
9 Maxwells equations and electromagnetic waves 49
1 Previous equations we have met . . . . . . . . . . . . . . . . . 49
4
-
8/7/2019 physik_iii
5/58
2 Maxwells Equation . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . 504 Energy transport in electromagnetic waves . . . . . . . . . . . 515 Lorentz transformations of electromagnetic waves . . . . . . . 52
10 Electric and Magnetic Fields in Mattter 531 Dielectric Materials . . . . . . . . . . . . . . . . . . . . . . . . . 532 electric dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.1 Torque and forces of electric dipoles in eletric fields . . 543 Atomic and molecular dipoles . . . . . . . . . . . . . . . . . . 55
3.1 Induced dipole moments . . . . . . . . . . . . . . . . . 553.2 Permanent dipole moments . . . . . . . . . . . . . . . . 55
4 Electric fields from polarized matter . . . . . . . . . . . . . . . 55
11 Magnetic Phenoma in Matter 56
12 Dielectrics and Refraction 57
Bibliography 58
5
-
8/7/2019 physik_iii
6/58
Contents
Abstract
This is going to be a summary of Prof Simon Lillys lecture "PhysicsIII - Electricity and Magnetism" in order to look up the basic facts andformulas quickly. Its intention is not to replace the script from 2005[1] which is very close to Prof. Lillys notes and online available1.But of course this summary contains the basic deductions necessary tounterstand the material.
In addition to the summarized lecture notes, I will add importantthings from the exercises (but give me some time to do this [and theexercises]).
Note that this summary is very incomplete at the moment. I hopethat I can present something like a final version about 1-2 weeksafter the end of the semester.
Please email2 me any errors in the sections which are not under con-struction.
1http://mitschriften.amiv.ethz.ch/[email protected] oder [email protected]
6
http://mitschriften.amiv.ethz.ch/main.php?page=3mailto:[email protected]:[email protected]:[email protected]:[email protected]://mitschriften.amiv.ethz.ch/main.php?page=3 -
8/7/2019 physik_iii
7/58
Chapter 1
Basics
1 Literature
The lecture is based on the "Purcell" [5]. Prof. Lilly recommended also the"Jackson" [2] and the "Knzig" [3]. All references are listed in the bibliogra-phy at the end of this document.
2 Constants and Units
2.1 Constants
electric permittivity 0 =1
0c20
= 8.854187 . . . 1012 AsVm
(1.1)
magnetic permeability 0 = 4107
= 1.25663706144 . . . 106 Hm
(1.2)
elementary charge e = 1.60217733 . . . 1019C (1.3)mass of electron me = 9.1093826 1031kg (1.4)
mass of proton mp = 1.67262171 1027kg (1.5)Boltzmanns constant k = 1.3806504(24) 1023 J
K(1.6)
= 8.617343(15) 105 eVK
(1.7)
7
-
8/7/2019 physik_iii
8/58
1. Basics
2.2 Units
1 C = 1A s (Coulomb) (1.8)
1 V = 1J
C= 1
W
A= 1
kg m2
s3 A(Volt) (1.9)
1 F = 1C
V= 1
A2 s4
kg m2(Farad) (1.10)
1 T = 1Vs
m2= 1
N
Am= 1
Wb
m2= 1
kg
As2= 104Gs (Tesla) (1.11)
1 H = 1Vs
A = 1 s (Henry) (1.12)
3 Terms
We use some special terms and abbreviations, these things are declared here.
EMF or emf is short for electro motive force. This is caused by mov-ing something through an electric or magnetic field, as defined in thecorresponding context.
AC stands for Alternating Current and means that we apply a currentthat changes periodically with time, in our cases sinusoidally.
DC stands for Direct Current and is used for currents which do notvary periodacally with time, in most cases a DC current is constant.DC currents can be damped, in this cases they are not constant any-more.
4 Concept of charge
two types of charge, named + and -
different types attract, same types repel each other
the total electric charge of an isolated system i
qi is constant
charge is relativistically invariant
charge in nature is quantized in units of the elementary charge e
8
-
8/7/2019 physik_iii
9/58
Chapter 2
Electrostatics
1 Force
1.1 Coulombs Law
The force between two charges acts on the shortes connection (vector #r 12)between the two charges and is described by Coulombs Law:
#
F =q1 q2
40r12#r 12 (2.1)
Additivity
The force on a charge x which located next to i other charges is the sum ofall forces.
#
Fx = i=x
1
40
qxqir2xi
#r xi (2.2)
1.2 Energy of a charge
If we bring a charge q1 next to another charge q2, we must do work on it.Therefore the charge has a stored energy
W1 =
r12
#
F 12 d#s = W2 (2.3)
9
-
8/7/2019 physik_iii
10/58
2. Electrostatics
Example: charges in NaCl
To visualize the superposition priniciple, one can calculate the total enery ofan ion in a NaCl-crystal with lattice constant a :
Wi =1
2
e2
40
6
a+
122a 8
3a+
6
2a ...
=
1
2
1
40
1.748e2a
(2.4)
If we sum up all the N ions, we get
W =1
2
1
40
1.748Ne2
a
(2.5)
2 Eletric field
The electric field is physical quantity, which is independent of the samplecharge.
#
E =
#
F
q(2.6)
#
E =1
40
all space
(x, y, z)r2
#r dxdydz (2.7)
2.1 Charge density
The charge density is a helping quantity to get the (differential) charge per(differential) volume.
=dF
dq(2.8)
(2.9)
It can be expressed in terms of the potential and the electric field:
= 0( #)2 (2.10) = 0
#E (2.11)
10
-
8/7/2019 physik_iii
11/58
2.2 Curl of the electric field
Because the eletric field is conservative, the curl vanishes.
rot#
E = rot( #) = 0 (2.12)
As a consequence from vector analysis, every closed path integral is 0.
3 Flux
The flux is defined as
d := E d #a :=
surface
E d #a =E d #a (2.13)
(2.14)
The flux through a closed surface generated by charges outside of this sur-face is = 0.
3.1 Flux and E-Fields of various charge distributions
spherical uniformly charged symmetric sphere
With the total charge qs and the surface charge density =qs
4r2swe get
inside: = 4r2E(r) =1
0
r0
dV E = 0 (2.15)
outside: = 4R2E(R) =qs0 E(R) = qs
4R20(2.16)
spherical uniformly charged cylinder
tba
11
-
8/7/2019 physik_iii
12/58
2. Electrostatics
infinite line charge
With the total charge ql and the line charge denisity =qll we get
= 2rl E(r) = l0 E(r) =
2r0(2.17)
infinite sheet charge
Total and surface charge density like before, it follows:
= 2r2 E(l) = r2
0 E(l) =
20(2.18)
4 Gauss Law
Gauss law relates the flux to the enclosed charge(s).
4.1 integral form
=S
#
E d #a =1
0
dV =
i
qi
0(2.19)
4.2 differential form
div E =
0(2.20)
12
-
8/7/2019 physik_iii
13/58
5 Potential
The electric potential is defined for every point of the room. It indicates theability to perform work.
AB = B
A
#
E d #s [] = 1V (Volt) (2.21)
usual convention : () = 0 (see def. of pot.) (2.22)
E = #
(2.23)
5.1 Definiton of potential 0
Potential can be only expressed in difference to another point. To make iteasy, it is a usual convention to set the potential in equal to zero.
() = limr (r) := 0 (2.24)
5.2 Potential of radial-symmetrical charge distribution
The potential for radial-symmetrical distributed charge (e.g. charged sphere)is
=1
40
rd #r (2.25)
5.3 Potential of multiple charges
The potential of multiply charges is just the superposition of the seperatepotentials.
(x, y, z) =1
40
N
i=1
qir
=1
40
V
(x, y, z)r
d #x (2.26)
13
-
8/7/2019 physik_iii
14/58
2. Electrostatics
5.4 Potential of a spherical shell
The potential of a charged spherical shell with radius R and charge Q isgiven by
(r) =1
40
Q
r(r > R) (r) =
1
40
Q
R(r < R) (2.27)
Inside the shell the potential doesnt change because there is no electric field.
5.5 Potential of an electric dipole
For an electric dipole made out of two charges |q1| = +q and |q2| = q at#x 1 = (0,0,
l2 ) and
#x 2 = (0,0, l2 ) the potential is given by
(r, , ) =1
40
p cos()
r2p = lq (2.28)
p is called the dipole moment, is the angle between the z-axis and #r .With in spherical coordinates we obtain the following expressions for thecomponents of the electric field:
Er = r
E = 1r sin()
E = 1r
(2.29)
6 Energy of an electric field
W =02
entire field
E2 dV (2.30)
enery density =0E2
2(2.31)
14
-
8/7/2019 physik_iii
15/58
Chapter 3
Conductors
1 Conductors and Insulators
The mobility in conductors is about 1020 higher than in insulators. Be-cause of this mobility charges in the conductor can rearange and adopt toexternally applied eletric fields. Therefore the eletric field within vanishes:
inside:#
E = 0 (3.1)
surface: = k = const.#
E surface (3.2)
2 General Electrostatic Problem
The general electrostatic problem is, that we a system of conductors withsome charge on it and want to determine the electric field arround them.We can distinguish these 3 cases:
1. k is defined ( Dirichlet problem)
2. Qk is defined ( Neumann boundary conditions)
3. a mix of the first two situations, which is not overdefined
3 Uniqueness Theorem
For a given situation, there is at most one solution.
This holds in our contex for the general electrostatic problem, in which wewant to determine the eletric field arround one ore more conductors.
15
-
8/7/2019 physik_iii
16/58
3. Conductors
4 Potential and Field inside an empty cavity in a closed
conductor
Inside a cavity which is fully enclosed by a conductore we have no E-fieldinside and a constant potential on the surface:
#
E = const = const on the surface (3.3)
solution: 2 = 0 = const in the cavity (3.4)
5 Mirror principle
We can use the uniqueness theorem to determine the influence of a chargebrought next to a conductor.
[tba: nice image of the mirror principle and the electric field]
Consider a charged in height h over an infinte sheet conductor:
Ez,q+ =1
40
q(r2 + h2)
cos Ez,q =1
40
q(r2 + h2)
cos (3.5)
cos = h(r2 + h2 Ez = 120 qh(r2 + h2)3/2 (3.6)
It follows the surface charge distribution and the total charge on the sheet,which is q as expected:
= 0Ez =1
2
qh(r2 + h2)3/2
Qs =
0
2rd #r = q (3.7)
Because of that, its called the mirror principle: Instead of thinking of theconductor and one charge, we think of two charges (see picture). The elet-
ric field at the location of the conduction is the same and because of theuniqueness theorem we solved the problem.
If the sheet is not infinite, the charges distribute like this: [tba: picture ofcharge distribution of not infinite sheet charge]
6 conducting sphere in external field
tba
16
-
8/7/2019 physik_iii
17/58
7 Capicatance and capacitors
7.1 Definition of capicatance
C =Q
U(3.8)
[C] = 1F = 1C
V (3.9)
7.2 Types of capacitors
Plate capicator:
E =F
d E = 1 2
d(3.10)
= 0E (3.11) C = A0
d(3.12)
(3.13)
Spherical capicator out of two spheres
C =Q
U=
QQ
40r1 Q
40r2
=40r1r2
r1
r2(3.14)
Spherical capicator out of one sphere
C = 40r1 (3.15)
A single sphere as a capacitor is the limit of the spherical capicator with oneshell at r2 = .
17
-
8/7/2019 physik_iii
18/58
3. Conductors
Cylindrical capicator
C =20l
ln r1r2(3.16)
7.3 Energy stored in a capicator
E =1
C
Q0
q dq =Q2
2C
U= QC=Q
2=
C2
2(3.17)
7.4 Systems of capicators
If we have a system of conductors, its helpful to introduce a capicatance for
each two conductors: Defining 1, 2, 3 defines#
E throughout the system(by uniqueness), so Q1, Q2, Q3 are determined (and vice versa). We canimagine various states:
State I: 2 = 3 = 0
all charges are proportional to 1
Q1 = C111 Q2 = C211 Q3 = C311 (3.18)
State II: 1 = 3 = 0
Q1 = C122 Q2 = C222 Q3 = C322 (3.19)
State III: 1 = 2 = 0
Q1 = C133 Q2 = C233 Q3 = C333 (3.20)
By superposition, general solutions are a sum of these states I, II, III
Q1 = C111 + C122 + C133 (3.21)
Q2 = C211 + C222 + C233 (3.22)
Q3 = C311 + C322 + C333 (3.23)
(3.24)
which we can write as
Qi =3
j=1
Cijj i = 1, 2, 3 (3.25)
where Cij is a matrix.
18
-
8/7/2019 physik_iii
19/58
Chapter 4
Currents
1 Electric currents
An electric current are (net-)charges in motion. In general, n charges of value
q are passing through an area#
A with an average speed #v , the current isdefined as
1.1 Definition of the current
I#A = nq#
A #v (4.1)
With the current density#
J the current can be calculated by
I =A
#
J d #a (4.2)
Current density
We define the current density
#
J = i niqi
#
v i [J] =G
m2 (4.3)
Closed surface
If we have a closed surface, we get with Gausss law
Is =A
#
J d #a =V
div#
J dv (4.4)
19
-
8/7/2019 physik_iii
20/58
4. Currents
If#
J is time-independent, no charge moves:
div#
J = 0 #J = const (4.5)
2 Ohms law and Resistance
The motion of charges is proportional to the electric field:
#
J = #
E (4.6)
2.1 Resistance and conductivity
is called the conductivity of the material.
=
#
J#
E[] =
1
m
= 1A
V
m
(4.7)
The inverse of the conductivity is the resistivity.
=1
[] = 1m (4.8)
Dont mix up resistivity with resistance, the resistivity is specific for eachplace. A resistance can be deteremined/measured between two points, it is
depends on the resistivity of the material between the two points.
2.2 Definiton of the resistance
We define the resistance
R =U
I[R] = 1 = 1
V
A(4.9)
20
-
8/7/2019 physik_iii
21/58
3 Resistance for various components
3.1 Discrete component
If we consider a discrete component of length l and with cross-section areaA, we get
I =
#
J d #a = AJ U =
#
E d #s = El (4.10)
R =
El
AJ
=L
A
=L
A
(4.11)
3.2 Spherical conductors
For two spherical conductors with radius a and b with a material inbetweenwith constant resistivity we get [4]
dR(r) = dr
4r2 R =
ba
dr
4r2=
4(
1
a 1
b) (4.12)
3.3 Cylindrical conductiors
For two cylindrical conductors with radius a and b and length (materialinbetween with constant resistivity as before) we get [4]
dR(r) = dr
2rl R =
ba
dr
2rl=
2lln(
b
a) (4.13)
4 Resistances in circuits
4.1 Parallel circuits
[tba: image of parallel]
Rres = i
Ri (4.14)
21
-
8/7/2019 physik_iii
22/58
4. Currents
4.2 Series circuits
[tba: image of series circuit]
1
Rres=
i
1
Ri(4.15)
4.3 Combined series and parallel circuits
Any combination of parallel and series circuits is imaginable. We reduceany complicated circuit into a combination of series and parallel circuits. Itholds:
1. Ohms law for each component
For each component
Ri =UiIi
(4.16)
2. Kirchhoffs law
The sum of all currents arriving at one node vanishes:
i
Iin = 0 (4.17)
3. Voltage through every loop
The voltage through every closed loop in the circuit is 0:
i
Vloop = 0 (4.18)
22
-
8/7/2019 physik_iii
23/58
5 Energy dissipation
The energy dissipated when a charge Q moves through a potential V is
E = VQ [E] = 1J = 1 Joule (4.19)
P = V I =V2
R= I2R [P] = 1W = 1 Watt (4.20)
6 Sources of electromotive force: Voltaic cellEnergy can come from a chemical reaction in the voltaic cell, imagine achemical reaction as a voltage source.
I =
R + RiV = IRi (4.21)
energy per unit charge emf= (4.22)(4.23)
Some energy is released as heat, therefore Ri and V as the available energy.
7 Circuits with capicators
[tba: small image with one circuit containing one capacitor, one resistor andone switch]
If we have a circuit with only one charged capacitor and one resistor, thecurrent flows and Q and V in the capacitor goes exponentially down.
V =Q
CI =
I
R= dQ
dt(4.24)
dQ
dt =Q
CR integrate: lnQ = t
RC + C (4.25)
Q = Q0e tRC I = dQdt
=V0R
etRC (4.26)
(4.27)
as an exercise one can show: (4.28)
0
V Idt =CV20
2=
Q202C
=Q0V0
2= energy stored at start (4.29)
23
-
8/7/2019 physik_iii
24/58
Chapter 5
Fields of moving charges
1 General force on a moving charge
#
F = q#
E + q #v #B (5.1)
2 Charge invariance
Charge is relativistically invariant.
This is becuse of the number of charges doesnt change in a system acceler-ated to relativistic velocities, because we have the same charge at relativisticvelocities. Consider a 2 H+2 - and a
4 He+-Ion. The have both 2 protons, 2neutrons and 1 electron but different masses. Although the mass differenceis measurable, there is no charge difference.
3 Fields of moving charges
4 Electric field for an accelarated charge
For a charge moving with constand velocity, we have an E-field that is radialform where the particle is.
If the particle accelerates or decelerates, the E-field for the particle is radialfrom where the particle would have been. Outside the sphere of radiusr = ct the field doesnt know that the charge has moved as the theorems ofspecial relativity state this.
[drawing from page 26]
24
-
8/7/2019 physik_iii
25/58
Energy in the spite
UE =
0E2
2dV dv = 2R2 sin dR d (5.2)
=0q
2a2
2(40)2c4c2[
3
4] UE =
q2a2
120c3(5.3)
Force on moving charges
Ex = Ex dpxdt = Exq = Exq (5.4)Ey = Ey
dpxdt
=Ey
(5.5)
result:
E =qa sin
40Rc2(5.6)
5 Forces between moving charges - the magnetic force
The force between a current in a conductor (i.e. many moving charges ofdensity lambda with a velocity vI result in a current I = v) and a stationarycharge moved perpendicular to the direction of the flowing current is
F =1
20
Iqv
rc2(5.7)
This force is perpendicular to the movement of the charge and perpendicularto the direction of the flowing current.
For the derivation, see [1]. This force is called magnetic force. Its orientationcan be determined by using the right-hand-rule and for currents in wires bythe kork-screw rule.
25
-
8/7/2019 physik_iii
26/58
Chapter 6
Magnetic fields
1 Definition of magnetic field
1.1 Lorentz force
The magnetic field is defined by the force acting on a moving charge:
#
F = q#
E + q( #v x#
B ) (6.1)
This is called the Lorentz force law, the first term is the electric field we knowalready from chapter 2, the second part is the magnetic field.
This magnetic field comes from moving charges i.e. currents.
1.2 Definition of B-field
B =1
40
2I
rc2= 0
I
2r[B] = 1T (6.2)
with the magnetic permeability
0 = 10c2== 1.25663706144 . . . 106 Hm (6.3)
To determine the direction in space of the B-field, one can use the right-hand-rule and the right corkscrew rule.
B I B E (6.4)
[image of B field going into the paper]
The magnetic field B comes from currents and acts on currents.
26
-
8/7/2019 physik_iii
27/58
2 Force between two wires
I1 = n1qv1 I2 = n2qv2 (6.5)
B1 =0 I12d
B2 =0 I22d
(6.6)
force per unit length = IB F1 = 0 I1 I2l2d
= F2 (6.7)
ni is number density of charges.
3 Properties of the magnetic field
For an infinite wire the B-field arround it is
B =0 I
2r(6.8)
3.1 Amperes Law
If we have a current I passing throught a loop, we have
#
B d #s = 0 I = 0
S
Jd #a (6.9)
With Stokes theorem we can write
S
#
F d#
s = A
rot F d#
a rot#
B = 0#
J (6.10)
Loops enclosing no currents
In general it holds for any closed loop through which no current flows
#
B d #s = 0 (6.11)
27
-
8/7/2019 physik_iii
28/58
6. Magnetic fields
4 Magntic poles
Because magnetic field lines do not end, it follows that there is no magneticmonopole (in contrast to that the eletric charge is a monopole).
A
#
B d #a = 0 rot #B = 0 #J (6.12)
5 Uniqueness theorem
Given a#
J(x, y, z), there is unique#
B (x, y, z).
div#
B = 0 rot#
B = 0#
J (6.13)
6 Vector potential
6.1 Deduction
We search for a potential for B. To get this, we start with a potential calledA:
B = rot#
A (6.14)
It holds
div#
B = div(rot#
A) = 0 (6.15)
rot(rot#
A) = 0#
J (6.16)
For each component one can deduce for each component
2 Ax = 0 Jx Ax = 04
space
Jxr
dV (6.17)
The general formular follows with doing this for each component x, y, z.
28
-
8/7/2019 physik_iii
29/58
6.2 General formula
2 #A = 0 #J #A = 04
space
#
J
rdV (6.18)
rot#
A =#
B (6.19)
By the uniqueness theorem, we relate by this formula the-one-and-only Bwith A and therefore A is unique.
6.3 Application to a current carrying wire
d#
A =I04r
d#
l =I04r
#
J dV d#
B = rotd#
A (6.20)
d #B = rotd #A = I04
rot(1
rd
#
l ) (6.21)
=
I0
4d#
l
1
r=
I0
4d
#
l
#
r
r2(6.22)
This is called the Biot-Savart-Law.
Biot-Savart-law
For a conductor of the differential length dl carrying a current I we get adifferential B-field
d#
B = Id#
l
#
r (0
4r2) (6.23)
The integral form of this law is
#
B (r) =04
V
#
J(r)#r
|r|3dV (6.24)
The vector r points from the place where the current is to the place we wantthe expression for the B-field.
29
-
8/7/2019 physik_iii
30/58
6. Magnetic fields
6.4 General differentials
Out of the last derivations, we get the following expressions for the generaldifferentials:
d#
B =#
J #r 04r2
dV (6.25)
d#
E = #
r1
4r20dV (6.26)
7 Fields of rings and solenoids
[image section 5.4, B-field straight up]
We can apply the Biot-Savart-Law:
on axis: Bz =0
4(b2 + z2)
Idl cos =
04(b2 + z2)
(2bI) cos (6.27)
=0 Ib
2
2(b2 + z2)3/2(6.28)
at z = 0: Bz =0 I
2b(6.29)
Solenoid with wire turns
[picture of solenoid]
We have n turns of wire per meter and therefore
I = Inr d
sin
b
r= sin (6.30)
dBz =0
2(
r dnI
sin )
b2
r3=
0
2In sin d (6.31)
Bz = 02
nI
21
sin d =02
In(cos 1 cos 2) (6.32)
long solenoid: 1 = 0 ; 2 = 2 Bz = 0nI (6.33)
#
B d #s = 0 I (6.34)
calculated on axis; but Amperes Law also fine for all r within the solenoid
30
-
8/7/2019 physik_iii
31/58
8 Change in B across a current sheet
[picture of sheet charge]
Remember that we have change in#
E across a charge sheet (see...tba), so theE-fields are:
#
E 1 #E 2 = E = 0
E = 0
(6.35)
pressure on sheet = ( E1 + E22 ) energy density = 12 0E2 (6.36)
We define a surface current density
#
Jin #x direction [J] = 1A
m(6.37)
Now we consider the little loop l and Amperes law:
#
B d #s = (B1 B2)l = 0 #J l Bz = 0 JxB = 0 #J B = 0 (6.38)B = 0 (6.39)
8.1 Energy density
In general the energy density of the magnetic field is given by
Energy density in B-field = 120
B2 (6.40)
Derivation
We consider a sheet carrying a flowing current and want to determine thepressure on the sheet....[tba]
[tba: picture form page 33]
31
-
8/7/2019 physik_iii
32/58
6. Magnetic fields
With the charge density n and the velocity v we get
dF = B(y) (dybc)nq charge
v (6.41)
#
B d#
d s = b dB = 0 (b dynqv) current
dB = 0nqv dy (6.42)
dF = Bnqv dybc =bc
0B dB (6.43)
We get the total force:
F =
bc
0
B2
B1
B dB (6.44)
Therefore the pressure is
Pressure = P =F
bc=
1
20(B22 B21) =
1
2(B1 + B2)(B1 B2) = 1
2(B1 + B2)J
(6.45)
Considering p dV work if sheet moves to sheet.It follows that:
Energy density in B-field =1
20 B2
(6.46)
(Compare this to that of the E-field which is E =10
2 E2).
9 Hall effect
[tba: Picture form page 34]
If we consider a current flowing throught a conductor under the influenceof an E field. The moving charges are subject to the Lorentz force. In caseof a normal conductor (metals etc.) we have moving electrons moving toone side of the conductor until we have an equilibrium of the Lorentz forceon the moving charges and the electrostatic force of the already separatedcharges.
q#
E z + q(#v #B) = 0 (6.47)
This phenomen can be used to determine what charges are moving in aconductor. Its called the Hall effect (1879)
32
-
8/7/2019 physik_iii
33/58
10 How do E and B transform together?
[tba: picture form page 34]
10.1 Derivation
In the lab fram:
Ey = 0
Bz = 0 j = v00 (6.48)
v0 =v0 v1 v0v
c2= c
0 1 0
=
0restframe ofplates
0 (6.49)
0 is defined in the restframe and comes from v0 and 0 comes from the
transformation to FWe have the transformation of:
= 1 0 (6.50)
With this we can write
j = v0 = 1 0c0
1 0 = v0 v (6.51)
Ey = 0=
0 0
0= Ey Bz
00c(6.52)
= Ey cBz00c2
= Ey vBz(6.53)
Bz = 0 j = 0v0 0v = Bz c00Ey (6.54)
= Bz (00)1/2Ey (6.55)= Bz
cEy (6.56)
33
-
8/7/2019 physik_iii
34/58
6. Magnetic fields
10.2 General transformation
In general (directions as in the picture) we get:
Ex = Ex Ey = Ey cBz Ez = Ez + cBy (6.57)
Bx = Bx By = By +
cEz B
z = Bz
cEy (6.58)
This was first derived 1905 by Einstein, it was a key step in joining the theoryof E and B fields.
The general vectoriell relations are:
E = E E = E + c B
B = B B = B c1 E
Note, that if there is a frame with B = 0, we get
B = c1 E = c1 Ebut with B = B = 0 we even get
B =
c1E
e.g a moving charge
IfE = 0, then we getE = c B
te be inserted/corrected:
If there is any frame with
#
B = 0 (6.59)
E = EE = E (6.60)
B = 0 (6.61)#
B = 1c #E #B perp =
c
#
#E = c
#
E
(6.62)
If there is a force with#
E = 0:
#
E = c #B (6.63)
34
-
8/7/2019 physik_iii
35/58
Chapter 7
Magnetic Induction
1 Magnetic Induction
The motion of conductors through magnetic fields can produces emf.
[tba: picture from page 36]
1.1 Isolated conductor
We have an accumulation of charges at ends until
#
E int =#v
#
B Ey = vBz (7.1)
In the rest-frame of the conductor we have:
Ey = [Ey cBz] = vBz (7.2)Eint = Ey = vBz Eint = vBz (7.3)
1.2 Loop of wire
[tba: picture from page 36 below] If we consider a loop moved perpendicularto a B field, we have an inducted current in the loop. This comes from theforces f2 und f1 along the pieces of lenght l. We can consider a constantmagnetic field because one can prove that for an inhomegenous field we getthe same solution.
E =
BA
q( #v #B) d #s = qvBl (7.4)
35
-
8/7/2019 physik_iii
36/58
7. Magnetic Induction
2 EMF - Electro motive force
We define the electro motive force, short emf, as
=work
charge(7.5)
In case of an isolated conductor (in our case our loop), we get the expression
= vlB =
d
dt
(7.6)
With an inhomegenous field with B1 at one end and B2 at the second end ofthe conductor we get the same solution (proof in the lecture):
= vl(B1 B2) (7.7)
2.1 General emf (electro-motive force)
In general, we have for the emf
=
( #v #B ) d #s (7.8)
cc
3 Magnetic flux
We define the magnetic flux through a loop as
(t) :=S
#
B d #a (7.9)
At later time we have
(t + dt) =
S+ds
B d #s = (t) + d (7.10)
The difference of the magnetic flux goes away (or in) as emf.
36
-
8/7/2019 physik_iii
37/58
4 Faradays law of induction
Faradays law of induction is known as
Uind = ddt
#
B d#
A ddt
(7.11)
It follows (see derivation)
rot#
E = #
B
t(7.12)
A magnetic field with a change in time produces an electric whirl field.
The magnetic field is not completely defined by this term as we can add
some components with rot#
E = 0.
4.1 Derivation
We start with a fixed loop in charging B (remember moving loop) and lookat it in the rest frame of the loop:
[tba:picture from page 37]
= v(B2 B1)l (7.13)[tba: second picture below]
#
E = #v #B because #E = 0 (Einstein transformation) (7.14)
#
E d#
s = = lv(E1 E2) energy gain per unit charge as charge moves
= v(B1 B2)l = = d
dt
(7.15)
Static loop
If we have a static loop, we can derive with Stokes theorem.C
#
E d #s = ddt
S
#
B d #a rot #E = t
#
B (7.16)
rot #E = Bt
(7.17)
This is conform with rot#
E = 0 in electrostatics.
37
-
8/7/2019 physik_iii
38/58
7. Magnetic Induction
5 Lenz Law
The emf drives I around the loop and Lenzs Law gives us the direction In
the case B1 > B2,d
dtis negative.
In general we have
The emf produces currents which act always to reduce the changein .
Because of this we have to put in energy into our system to maintain aconstant current (E = I2R).
This law was first stated by Heinrich Lenz in 1834. It is nothing more thana special case of the principle of Le Chatelier:
"Any change in status quo prompts an opposing reaction in theresponding system."
6 Mutual inductance
[tba:picture] We consider the general case of two circuits at a small distance.A change in current in one circuit may produce a change in in the othercircuit and therefore produces emf.
21 =S2
#
B 1 d#a2 = kI1 emf= 21 = d21
dt= k dI1
dt(7.18)
We define the constant M = M21 as the mutual inductance, which has thefollowing units:
[M] = 1Vs
A= 1 s = 1 H = 1 Henry (7.19)
One can proof with Stokes theorem, the mutual inductances of two loopsare equal (see [Skript] for that):
M = M21 = M12 21 = 12 I1 = I2 (7.20)
38
-
8/7/2019 physik_iii
39/58
7 Self inductance
A change in the magnetic flux through a loop of wire causes a change in thecurrent flowing through the loop. We define the self-inductance via
=d
dt=: LI
t(7.21)
[L] = 1Vs
A= 1s = 1H = 1 Henry (7.22)
7.1 Circuits with self-inductance
[tba: picture p.39]
Charging the solenoid
We close the switch at t = 0.
0 L dIdt
= RI 0 = RI+ L dIdt
(7.23)
The solution of this DGL is
I =0R
(1 e RL l (7.24)
Decharging the solenoid without diode
If we open the switch, the solenoid will release a very high emf. This veryhigh emf can produce high voltages (can be much higher than the usedvoltages).
Decharging the solenoid with diode
To prevent this, one can add a diode in the circuit:
[tba: picture: circuit with diode]
As a result, the magnetic field goes down exponentially. We can calculate
39
-
8/7/2019 physik_iii
40/58
7. Magnetic Induction
that by integration the DGL with boundary values I(t0) = I0 and t = t0.
LdI
dt+ RI = 0 I = I0e RL (tt0) (7.25)
8 Energy in a magnetic field
A magnetic field can store energy. In case of a solenoid we can store theenergy by buildung up the magnetic field with a voltage and obtain thestored energy by disconnecting the voltage source.
We obtain the energy for a current loop or a solenoid in a circuit:
Emagnetic =LI20
2(7.26)
I0 is current flowing through the loop/solenoid before we release the en-ergy (e.g. opening a switch).
8.1 General term for any B-field
Emagnetic =1
0
V
B2 dV (7.27)
8.2 Derivation of the energy stored in a solenoid
We can derive this expression for the energy stored in magnetic field bysetting up a magnetic field inside a solenoid...
[tba: Herleitung]
9 Application: Magnetic field brake
In an exercise (8.2 [4]), we had a moving rod in a magnetic field, the ends ofthe rod are connected to a resistance R. The emf produced by the magnetic
40
-
8/7/2019 physik_iii
41/58
field decelerated the rod:
= ddt
= ddt
S
#
B d#
A = BL dxdt
= BLv (7.28)
I =
R= BLv
R F = BIL (7.29)
eq. of motion: 0 =dv
dt+
(BL)2
mRv v = v0e
(BL)2mR t (7.30)
41
-
8/7/2019 physik_iii
42/58
Chapter 8
Alternating Currents
1 Definitions for AC-circuits
1.1 Quality factor
Q =energy soted in system
rate of energy loss = radians befoe energy goes down to 1
e(8.1)
For
V = V0et cos(t)underdamped solution (8.2)
The energy is V2
et
.
dE
dt= 2E (8.3)
under-damped
Q =
2(8.4)
for near to cirital LR since = R2L
R2
4L2 1 1RC .
Q is related to a phase shift
V = V0etcos(t) I = C dVdt
(8.5)
I = CV0et sin(t + ) 1cos
= tan1
=
1
Q(8.6)
2 RCL circuit
[tba: picture from page 43]
42
-
8/7/2019 physik_iii
43/58
-
8/7/2019 physik_iii
44/58
8. Alternating Currents
With Kirchhoffs law:
0 cos(t) = LdI
dt+ RI(as before in DC case) (8.16)
tan = LR
= 1Q
I0 =0
(R2 + 2L2)(8.17)
The current I lags behind with a phase shift and Ihas a reduced amplitudeI0. In case of a DC current these formulars still hold, the simply go over intothe DC-forumulas.
[tba: graphic of I and 0]
5 RC-circuit
Similiar to the RL-circuit we get
tan =1
RCI0 =
0
(R2 + 1C
2)
(8.18)
6 RLC-circuit
VL = V1 V2 = L dIdt
VC = QC
(8.19)
I = I0 cos(t + ) VL = I0L sin(t + )(8.20)
VC = 1CIdt = I0
Csin(t + ) (8.21)
V = VL + VC = (L 1C
)I= sin(t + ) (8.22)
It is conform with
LI0 sin(t + ) forL = L 12C
(8.23)
or1
C= (L 1
C) (8.24)
44
-
8/7/2019 physik_iii
45/58
I0 =0
R2 + L 1C2
(8.25)
for fixed 0 adn we get max I0 (and zero phase lag) when
L =1
C max = 1
LC(8.26)
[graph: resonance curve, see p.43]
Power
Power I2 Power halfes when (L 1C
= R (8.27)
Expanding at about max peak leeaks to the half-peak power consumptionof
2
max=
R
maxL=
1
Q(8.28)
7 Application: Media player
changable resistor change in I, therefore change in volume put in asolenoid bass boost capacitore boost in high frequencies[summary of all 3 cases with circuits]
8 AC currents and voltages as complex numbers
We can represent AC currents and voltages as complex numbers. For thisthe current has to vary sinusoidally with time and the components (L, C, R)have to respond linear to the voltage.
ei = cos + i sin I = I0 cost + = Re(I0ei
eit) (8.29)
= complex number representating relative amp and phase relative to V
45
-
8/7/2019 physik_iii
46/58
8. Alternating Currents
It is easy to show that representation of sum of currents is equal to the sum
of their representations. With this we can analyze arbitrarly complicatedcircuits.
Usual rules (of circuits with R only):
1. At any junction:
Iin = 0 i
I0,i cost + i = 0 (8.30)
i
I0ei = 0 (8.31)
2. Arround any loop:
Voltage chips = em f (8.32)
for LR-circuit:
V = 0 cost (8.33)
I = I0 cost + (8.34)
tan = LR
(8.35)
I0 =0
R2 + 2E(8.36)
We can write
I = YV (all complex numbers) (8.37)
with Y =ei
R2 + 2E(admittance) Z =
1
Y(impedance) (8.38)
Now:
Y =1
RZ = R for pure R (8.39)
Y = iL Z = iL for pure L (8.40)Y = iC Z =
iC
for pure C (8.41)
Laws for addition
I = I1 + I2 Y = Y1 + Y2 for parallel circuit (8.42)V = V1 + V2 Z = Z1 + Z2 for series circuit (8.43)
This is the same as adding resistors in R-circuits.
46
-
8/7/2019 physik_iii
47/58
Example 1
[graphics marker 7.1]
I = YV = [YR + YC + YL]V (8.44)
= 0[1
R+ i(C 1
L)] (8.45)
Example 2
[graphics marker 7.2]
I =V
Z=
V
ZR + ZL + ZC=
V
R + i(L 1C )(8.46)
= arctan1
R(
1
CL) (8.47)
I0 =0
R2 + (L 1C )2 (8.48)
General for components in parallel
[graphics marker 7.3]
I0 = 0[1
R+ i(C +
1
L] = arctan R(C 1
L) (8.49)
9 Power consumptation in AC current
Instantaneous power consumptation = V I (8.50)
Power on average =< V I> (8.51)
47
-
8/7/2019 physik_iii
48/58
8. Alternating Currents
For pure resistor circuit:
< V I> =V20R
< cos2 t >=V202R
=I20 R
2(8.52)
V I = 0 I0 cost cos (t + ) (8.53)
= 0 I0[cos2 t cos (cost sint
cancels out
sin )] (8.54)
= 0 I0 < cos2 t > cos (8.55)
10 Some other definition
V = 0 cost :=
20 cost (8.56)
I = I0 cos (t + ) :=
2I0 cos (t + ) (8.57)
Power = 0 I0 12
cos := 0 I0 cos (8.58)
These second definitions are also possible but not used in the lecture. Theymay be found in some books.
48
-
8/7/2019 physik_iii
49/58
-
8/7/2019 physik_iii
50/58
9. Maxwells equations and electromagnetic waves
C
#
B d #s = 0 I S
rot#
B d #s =S
0#
J d #s (9.8)
I =dQ
dt=
d
dt(A) = A0
E
t(9.9)
rot B = 0 #J + 00 Et
(9.10)
Now: div(rot#
B ) = div0#
J + divE
t00 = 0
t+ 0
t= 0 (9.11)
0E
t : = displacement current (9.12)
2 Maxwells Equation
div#
E =
0Coulombs Law Gauss Law (9.13)
div#
B = 0 no magnetic monopoles (9.14)
rot#
E = dB
dt Faraday induciton (9.15)
rot#
B = 0#
J + 00E
tAmperes Law2 (9.16)
div#
J = t
continuity equation for charge (9.17)
(9.18)
These equations are called Maxwells equations, stated by Maxwell in 1865which was before the Theory of Relativity, the nature of matter was under-stood and before the link between light and electromagnetism was discov-ered.
3 Electromagnetic Waves
There are wave-like solutions to Maxwells equations in vacuum.
#
E = #z E0 sin(ky t) #B = #x E0 sin(ky t) (9.19) rot #E = #x kE0 sin(ky t) rot #B = #z kE0 sin(ky t) (9.20)
50
-
8/7/2019 physik_iii
51/58
In a complete empty space we get with Maxwells Euquations
rot#
E = #
B
trot
#
B =1
c2#
E
t(9.21)
Therefore Maxwells Euqations are forfilled with the following conditions:
c =100
=
kE0 = cB0 (9.22)
#
E#
B = 0 (9.23)
0E20
B200
= 0 (9.24)
#
v #
E #
B (9.25)
The directions of travel is#
E #BMaxwell postulated that light is an electric and magnetic wave.
4 Energy transport in electromagnetic waves
dE = (
0E2
2 +
1
20 B2
) 1
20E200=
12 0E
2
dV (9.26)
= 0E2 dV =
1
0B2 dV (9.27)
This energy is equally split between the E-field and the B-field.
average energy density = 0 < E2>=
1
0< B2 > (9.28)
=1
2
0E20 =
1
0
B20 (9.29)
energy transport per unit average = 0 < E2> c =
1
0< B2 > (9.30)
We introduce the Poynting Vector
#
S =c00
#
E #B (9.31)
This vector gives back the instantaneous energy transport per unit area tothe transport direction.
51
-
8/7/2019 physik_iii
52/58
9. Maxwells equations and electromagnetic waves
5 Lorentz transformations of electromagnetic waves
Ex = Ex Ey = Ey cBz Ez = Ez + cBy (9.32)
Bx = Bx By = By + c
1 Ez Bz = Bz c1 Ey (9.33)By simply calculating we can see
E B = Ex Bx + Ey By + EzBz (9.34)= Ex Bx +
2Ey cBzBy + c1Ez (9.35)
+ 2
Ez + cByBz c1
Ey (9.36)= Ex Bx +
2(1 2)EyBy + EzBz (9.37)= E B (9.38)
and
E 2 c2B 2 = E2x + 2Ey cBz2 + 2E2z + cBy2 (9.39)
c2B2x + 2By +
cEz
2
+ 2B2z
cEy
2
(9.40)
= E2 c2B2 (9.41)
0E2 1
0B2 = 0E2 1
0B2 (9.42)
If E and B are perpendicular in the frame F, then also in an other frameF. IfE2 = c2B2 in F, then also in F and an electromagnetic wave in F isan electromagnetic wave in F. In the frame of the wave let be Ey = E0,Bz = c1E0, then
Ey = E0 c
cE0 = E0(1 ) (9.43)
Bz = E0c c E0 = E0c 1 (9.44)
where
(1 ) =
1 1 +
So as 1, E 0 and B 0. This means, the frame that moves withlight speed in direction of the wave, the wave simply doesnt exist anymore.Therefore, the Lorentz transformations actually does change the amplitudesof the fields (and of course not the wave speed).
52
-
8/7/2019 physik_iii
53/58
Chapter 10
Electric and Magnetic Fieldsin Mattter
1 Dielectric Materials
When a material is introduced into a capacitor, the capacitance changes.
in vacuum: C =Q
V(10.1)
with material C = rC with > 1 (10.2)given V: Q = r Q (10.3)given Q: V = V
r E = E
r(10.4)
r medium
1 vacuum by definiton
1.00-1.01 typical gases
1.00059 air at 273 K and atmospheric presure
20-80 typical liquids
60 CH3OH
80 water at 293 K
2-10 typical solids
53
-
8/7/2019 physik_iii
54/58
10. Electric and Magnetic Fields in Mattter
2 electric dipoles
For two displaced charges:
#p = ql #r + (10.5)#p =
#r dV (10.6)
2.1 Torque and forces of electric dipoles in eletric fields
Torque
The torque on an electric dipole in an electric field is
#
N = #p #E (10.7)
Work to align the dipole
The work to align the dipole from the
state to the
state is
W = pE (10.8)
This is equal to the work necessary to align an arbitraly orientated dipole.
Potential energy
The potential energy of the dipole is given by
potential energy = #p #E (10.9)
Net force in external E-field
The net force on the dipole in an external#
E-field is
#
F =
#pEx#pEy#pEz
(10.10)
54
-
8/7/2019 physik_iii
55/58
3 Atomic and molecular dipoles
3.1 Induced dipole moments
Atoms and molecules can act as dipoles under influence of an external elec-
tric field. An eletric field#
E causes a dipole moment
#p = 40#
E =9
2a30 (10.11)
The constant is called the atomic polarizability and a0 the Bohr radius (de-duction of the formular for in quantum mechanics lectures). This dipolemoment yields in an dielectric constant r 2
Estimation of
| #E int | e40a20
z
a Eext
Eint Ee
40a2(10.12)
p = ez 40Ea30 (10.13)
3.2 Permanent dipole moments
Asymmetric molecules can have a not-vanishing #p also if there is no exter-
nal#
E because of the differently distributed charge in the molecule (partialcharges). As a result materials with polar molecules can have much biggerdieleletric constants, e.g. in case of H2O r 80.
4 Electric fields from polarized matter
Material emposed of aligned dipoles of average strength #p and a numberdensity N have a total dipole moment of
#
P tot =#p NdV =
#
P dV (10.14)
with the density of polarization#
P .
55
-
8/7/2019 physik_iii
56/58
Chapter 11
Magnetic Phenoma in Matter
56
-
8/7/2019 physik_iii
57/58
Chapter 12
Dielectrics and Refraction
57
-
8/7/2019 physik_iii
58/58
Bibliography
[1] Raphael Honegger. Physics III - Script of the lecture. VMP, 2005.
[2] John David Jackson. Classical Electrodynamics. Wiley-VCH, 1998.
[3] Werner Knzig. Elektrizitt und Magnetismus. vdf Hochschulverlag AG,2003.
[4] Prof. Dr. Simon Lilly, editor. Physics III - Exercises, HS 2008.
[5] Edward M. Purcell. Electricity and Magnetism (Berkeley Physics Course,Vol.2). McGraw Hill Higher Education, 1984.